Author: Mike ShulmanFormat: MarkdownCreated [[extranatural transformation]] by moving the relevant information from [[dinatural transformation]] and adding the definition. Disagreements are welcome, but I feel that since dinaturals that aren't extranatural are so rare and harder to deal with and understand, extranaturals merit their own page.

Created extranatural transformation by moving the relevant information from dinatural transformation and adding the definition. Disagreements are welcome, but I feel that since dinaturals that aren't extranatural are so rare and harder to deal with and understand, extranaturals merit their own page.

Author: YaronFormat: MarkdownItexI have a (perhaps silly) question regarding the entry [[extranatural transformation]]. After Lemmas 1-3, it is stated that "In fact, these lemmas essentially capture “all possible” ways in which extranatural transformations can be composed. The general statement, which is obtained by combining these, is that if the graphs representing the two transformations can be composed without creating any closed loops, then the transformations can be composed, and the resulting graph is the composite of the individual graphs."
However, I don't understand how, for example, the general version of yanking, where the domain of $G$ has any odd number of factors (Prop. 1 of Eilenberg-Kelly) follows by combining Lemmas 1-3 as stated. To really become the "general building blocks," shouldn't Lemma 1-3 be stated with any number of even (Lemma 1,2) or odd (Lemma 3) factors for the relevant transformations, as in Props. 1,2,2* of E-K?

I have a (perhaps silly) question regarding the entry extranatural transformation. After Lemmas 1-3, it is stated that “In fact, these lemmas essentially capture “all possible” ways in which extranatural transformations can be composed. The general statement, which is obtained by combining these, is that if the graphs representing the two transformations can be composed without creating any closed loops, then the transformations can be composed, and the resulting graph is the composite of the individual graphs.”

However, I don’t understand how, for example, the general version of yanking, where the domain of GG has any odd number of factors (Prop. 1 of Eilenberg-Kelly) follows by combining Lemmas 1-3 as stated. To really become the “general building blocks,” shouldn’t Lemma 1-3 be stated with any number of even (Lemma 1,2) or odd (Lemma 3) factors for the relevant transformations, as in Props. 1,2,2* of E-K?

Author: Mike ShulmanFormat: MarkdownItexNice, thanks!
I wonder whether it would be worth stating the theorems in the simpler $n=1$ case first, to help the reader's intuition, before the general one with all the indices?

Nice, thanks!

I wonder whether it would be worth stating the theorems in the simpler n=1n=1 case first, to help the reader’s intuition, before the general one with all the indices?